In cosmology, a basic explanation of the observed concentration of mass

in singular structures is provided by the Zeldovich approximation, which

takes the form of free-streaming flow for perturbations of a uniform

Einstein-de Sitter universe in co-moving coordinates. The adhesion

model suppresses multi-streaming by introducing viscosity. We study

mass flow in this model by analysis of Lagrangian advection in the

zero-viscosity limit. Under mild conditions, we show that a unique

limiting Lagrangian semi-flow exists. Limiting particle paths stick

together after collision and are characterized uniquely by a

differential inclusion. The absolutely continuous part of the mass

measure satisfies a Monge-Ampère equation related to convexification of

the free-streaming velocity potential.

The use of Monge-Ampère equations and optimal transport theory for the

reconstruction of inverse Lagrangian maps in cosmology was introduced in

work of Brenier and Frisch et al (2003). We show that the singular part

of the mass measure can differ from the Alexandrov solution to the

Monge-Ampère equation, however, when flows along singular structures

merge, as shown by analysis of a 2D Riemann problem. In a neighborhood

of merging singular structures in our examples, we show that

reconstruction yielding a monotone Lagrangian map cannot be exact a.e.,

even off of the singularities themselves.